1. Field of the Invention
This invention provides a computer-aided migration routine that is efficient and economical of computer time. The method is effective in the presence of moderate vertical and lateral velocity gradients and is capable of imaging steep dips and diving waves.
2. Discussion of the Prior Art
Although the art of seismic exploration is very well known, it will be briefly reviewed to provide definitions of technical terms to be referenced herein.
An acoustic source of any desired type such as, by way of example but not by way of limitation, a vibrator, an explosive charge, an air or gas gun, or an earth impactor, is triggered to propagate a wavefield radially from a source location. The wavefield insonifies subsurface earth formations whence it is reflected therefrom to return to the surface. The mechanical earth motions due to the reflected wavefield are detected as electrical signals by an array of seismic receivers or receiver groups distributed at preselected spaced-apart group intervals, at or near the surface of the earth, along a designated line of survey.
The mechanical motions detected by the receivers are converted to electrical, digital or optical signals which are transmitted over ethereal, electrical or optical data-transmission links to a multi-channel recording device. Usually, each receiver is coupled to a dedicated recording channel. An array may encompass many tens or hundreds of receivers which are coupled by the transmission link to a corresponding number of data-recording channels. To reduce the need for an excessive number of individual data transmission lines between the receivers and the recording channels, the receivers share a relatively few common transmission lines and the signals from each receiver are multiplexed into the appropriate data-recording channels by any convenient well-known means.
In operation, the selected source type successively occupies a plurality of source locations along the line of survey, launching a wavefield at each location. After each launch, the source is advanced along the line by a multiple of the receiver-spacing interval. At the same time, the receiver array is advanced along the line of survey by a corresponding distance.
The signals detected by the receivers and recorded on the respective data recording channels comprise an original set of raw seismic reflection data signals. The raw seismic data signals may be combined into gathers, processed, migrated and formatted into displays that image the subsurface earth layers along the line or area of survey under study.
The original analog seismic signals are usually digitized prior to recording for convenience in later processing. Following selected data processing routines, the processed digital data signals from each channel are then converted back to analog form for physical display as a sequence of time-scale oscillogram that clearly depict the configuration of the subsurface earth layers under study.
The original raw seismic reflection signal data sets are gathered solely to provide a picture of the subsurface of the earth for the purpose of exploiting the natural resources thereof for the benefit of humankind. Between the activities of data-gathering and data-displaying the seismic data signals may be subjected to the mathematical equivalents of filtering and other processes to remove noise, to improve the clarity of inter-layer bedding resolution, to reposition, as by migration, out-of-plane reflections and to perform other selected ministerial options.
It is recognized in the geophysical industry that seismic data-processing routines, which may be digital, exist to enhance the geophysical presentation and interpretation of subsurface geology as derived from the raw gathered seismic reflection signals. The seismic reflection signals are not gathered to solve some abstract mathematical formula.
Throughout this disclosure and wherever encountered, the simple phrase "data set" is a short-hand term that means "seismic reflection signal data set".
By definition, migration, as applied to the seismic art, is an inversion operation involving rearrangement of seismic data information sets so that the reflections and diffractions are plotted at their true locations. The need for this arises since the variable velocities and dipping horizons cause the elements of the data sets to be recorded at surface locations that are different from the true subsurface positions. The process is sometimes referred to as imaging. Migration can be accomplished by migrating along diffraction curves, by numerical finite differences or phase shift downward-continuation of the wavefield, and by equivalent operations in the frequency-wavenumber (f-k), frequency-offset (.omega.-x) or other domains.
Migration methods in the wavenumber-frequency domain have been used extensively in the past to accomplish both prestack and poststack migration of seismic data. Those methods that extrapolate in depth typically use a phase shift operator whose argument is the depth step multiplied by the z-wavenumber. The main advantage of those methods is their ability to image extremely steep dips and even overturned reflection events. Their main drawback, however, is their inability to correctly handle lateral velocity variations. In order to handle such variations more correctly, several extensions to the phase shift method have been introduced. These methods include interpolation based techniques and techniques involving perturbation series expansions, such as the generalized f-k method of Pai in Generalized F-K Migration In Arbitrarily Varying Media, (1988), Geophysics, 53, pp 1547-1555. Interpolation methods can be costly because they require a Fourier transform at every depth step for each reference velocity used. If the number of reference velocities is too small, inaccuracies associated with the interpolation process arise.
In contrast, if velocity variations are not too severe, the generalized f-k approach may be the cheaper in principle because only one application of the slowness-squared operator is required at each depth step to first order in perturbation theory. That is equivalent to doing one forward and one inverse Fourier transform. However, the perturbation series approach has the drawback that the expansion is in terms of (s.sub.0 .omega..sup.2 --k.sup.2).sup.-1/2 arising from the free-space Green's function. Here s.sub.0 is the zeroth Fourier component of the slowness squared and k is the spatial wavenumber. At steep dips, those factors diverge and hence the perturbation expansion becomes unstable.
Other methods for accomplishing depth migration in the presence of lateral velocity variations are known. The method described by D. Kosloff and E. Baysai, 1983 in Migration With The Full Acoustic Wave Equation, Geophysics, v. 48, pp 677-687, used a fourth-order Runge-Kutta method to solve the two-way wave equation. This method is carried out in the .omega.-x domain, however, the diving waves cannot be imaged. Further, stability restrictions because of evanescent energy necessarily limits the maximum dip that can be imaged using this technique in low-velocity areas.
Another alternate method for accomplishing steep-dip and diving-wave migration is Gaussian Beam Migration as taught be N. Hill in Geophysics, 1990, v. 55, pp 1416-1428. In that method the migration is not carried out in either the .omega.-x or f-k domains, but is done locally in a reference frame that is relative to the ray path. The method relies on an asymptotic solution to the wave equation however, and has difficulty imaging in areas where steep velocity gradients exist.
There is a need for an alternate approach to the generalized f-k migration that does not rely explicitly on any perturbation series expansion to thus avoid the steep-dip instability present in the prior-art formulations. The method should be able to image steep dips and diving waves if the lateral velocity variations are not too severe. The method should reduce to simple phase shift migration for velocity fields that are a function of depth only.